February 2, 2008 16:46 WSPC/INSTRUCTION FILE Lih-ICFP07 International JournalofModernPhysicsA (cid:13)c WorldScientificPublishingCompany 8 0 0 DECAY SPECTRUM OF K+ → e+νeγ 2 n a J C.H.CHEN1,C.Q.GENG2 andC.C.LIH3 7 1Department of Physics, National Cheng-Kung University,Tainan 701, Taiwan 2Department of Physics, National Tsing-HuaUniversity,Hsinchu 300, Taiwan ] 3General Education Center,Tzu-Chi College of Technology, Hualien 970, Taiwan h p - TheformfactorsoftheK+→γtransitionarestudiedinthelight-frontquarkmodeland p e chiral perturbation theory of O(p6). The decay spectrum of K+ → e+νeγ, dominated h bythestructuredependent contribution, isillustratedinbothmodels. [ 1 It is known that the decay of K+ e+ν γ receives two types of contributions: e v “inner bremsstrahlung” (IB) and “st→ructure-dependent” (SD) 1,2. The former is 4 7 helicity suppressed and contains the electromagnetic coupling constant α, while 0 the latter gives the dominant contribution to the decay rate as it is free of the 1 helicity suppression. In the standard model (SM), the decay amplitude of the SD . 1 partinvolves vectorand axial-vectorhadronic currents,which canbe parametrized 0 interms ofthe vectorformfactor F andaxial-vectorformfactorF ,respectively. 8 V A 0 However, the experimental determinations on these form factors are poorly given : andmodel-dependent3,4,5.Inparticular,theexperimentalresultsonthedecayrate v i ofK+ e+νeγinRef.3,4,5werebasedontheassumptionofFV andFAbeingsome X constan→tvaluesinthechiralperturbationtheory(ChPT)atO(p4)6.Intheongoing r a data analysisofthe E949experimentatBNL, moreprecisionmeasurementsonthe decay of K+ e+ν γ are expected 7 andthus, the model-independent extractions e → of the SD form factors are possible. Theoretical calculations of F and F in the V A K+ γ transition have been previously done in the ChPT at O(p4) 6 and O(p6) 8,9 a→s ell as the light-frontquark model (LFQM) 10. However,the results have not been fully applied to the decay of K+ e+ν γ yet. e → 11 In this talk, we will present our recent results on the transition form factors of K+ γ in the ChPT of O(p6) and light-front quark model (LFQM). We will → show the spectrum of the differential decay branching ratio of K+ e+ν γ as a e → function of x=2E /m . γ K WestartwiththeamplitudeofthedecayK+ e+ν γintheSM,givenby2,6,12 e → M = M +M , IB SD G G M = ie Fsinθ F m ǫ∗Kα, M = ie Fsinθ ǫ∗L Hµν, (1) IB √2 c K e α SD − √2 c µ ν where Kα = u¯(p )(1 + γ ) pαK 2pαe+q6γα v(p ), L = u¯(p )γ (1 γ )v(p ), ν 5 pK·q − 2pe·q e ν ν ν − 5 e (cid:16) (cid:17) 1 February 2, 2008 16:46 WSPC/INSTRUCTION FILE Lih-ICFP07 2 Chen, Geng and Lih Hµν = FA ( gµνp q+pµqν)+iFV ǫµναβq p ǫ is the photon polarization mK − K · K mK α Kβ α vector, p , p , p , and q are the four-momenta of K+, ν , e+, and γ, and F and K ν e e K F are the K meson decay constant and the axial-vector (vector) form factor A(V) corresponding to the axial-vector (vector) part of the weak currents, respectively, defined by F 0s¯γµγ uK+(p ) = iF pµ, γ(q)u¯γµsK(p ) =ie V εµαβνǫ q p , h | 5 | K i − K K h | | K i m ∗α β ν K F γ(q)u¯γµγ sK(p ) =e A [(p q)ǫ∗µ (ǫ∗ p)qµ], (2) 5 K h | | i m · − · K with p = p q being the transfer momentum. We note that in Eq. (1) K IB − M is suppressed due to the small electron mass m . In the decay of K+ e+ν γ, e e → the form factors F in Eq. (2) are the analytic functions of p2 = (p q)2 in A,V K − the physical allowed region of m2 p2 m2 . The relation between the transfer e ≤ ≤ K momentum p2 and x is given by p2 =m2 (1 x). At O(p6) in the ChPT, one obtains tKhat−11 m 256 64 F (p2)= K 1 π2m2 Cr+256π2(m2 m2)Cr + π2p2Cr V 4√2π2F − 3 K 7 K − π 11 3 22 K(cid:26) 1 3 m2 7 m2 m2 m2ln η + m2 ln π +3m2 ln K −16π2(√2FK)2(cid:20)2 η µ2 ! 2 π (cid:18)µ2 (cid:19) K (cid:18) µ2 (cid:19) xm2 +(1 x)m2 x(1 x)p2 2 xm2 +(1 x)m2 x(1 x)p2 ln π − K − − dx − π − K − − µ2 Z (cid:18) (cid:19) (cid:2) (cid:3) xm2 +(1 x)m2 x(1 x)p2 2 xm2 +(1 x)m2 x(1 x)p2 ln η − K − − dx − η − K − − µ2 ! Z (cid:2) (cid:3) m2 4 m2 ln π dx , (3) − π µ2 Z (cid:18) (cid:19) (cid:21)(cid:27) 4√2m m F (p2)= K(Lr+Lr )+ K [142.65(m2 p2) 198.3] A F 9 10 6F3(2π)8 K − − K K m m2 m2 K (4Lr+7Lr+7Lr )m2 ln π +3(Lr+Lr )m2ln η −4√2FK3π2 ( 3 9 10 π (cid:18)m2ρ(cid:19) 9 10 η m2ρ! m2 +2(8Lr 4Lr+4Lr+7Lr+7Lr )m2 ln K 1− 2 3 9 10 K m2 (cid:18) ρ (cid:19)(cid:27) 4√2m K 2m2(18yr 2yr 6yr +2yr +3yr yr +6yr ) − 3F3 π 18− 81− 82 83 84− 85 103 K +2m2 (18(cid:8)yr +36yr 4yr 12yr +4yr +6yr +4yr 3yr K 17 18− 81− 82 83 84 85− 100 3 +6yr +12yr 6yr +3yr )+ (m2 p2)(2yr 4yr +yr ) , (4) 102 103− 104 109 2 K − 100− 109 110 (cid:27) where Cr, Lr and yr are the renormalized coupling constants. Note that the first i i i terms in Eqs. (3) and (4) correspond to F and F at O(p4) 6, respectively. V A February 2, 2008 16:46 WSPC/INSTRUCTION FILE Lih-ICFP07 Decay Spectrum of K+→e+νeγ 3 10 11 In the framework of the LFQM , we obtain dzd2k 1 2m Ak2Θ 1m +Bk2Θ F (p2)= 4m ⊥Φ z′,k2 u− ⊥ + s ⊥ , A K 2(2π)3 ⊥ 1 z′ 3 m2 +k2 3 m2+k2 Z − (cid:26) u ⊥ s ⊥ (cid:27) dzd2k (cid:0) (cid:1) 1 F (p2)= 8m ⊥Φ z′,k2 V K 2(2π)3 ⊥ 1 z′ Z − 2m z′(m (cid:0)m )k2(cid:1)Θ 1m +(1 z′)(m m )k2Θ u− s− u ⊥ s − s− u ⊥ , (5) 3 m2 +k2 − 3 m2+k2 (cid:26) u ⊥ s ⊥ (cid:27) 11 where the parameters and variables are defined in Ref. . The numerical values of F (p2) in the ChPT of O(p6) are plotted in Fig. 1. A,V Inthesefigures,wehavealsoincludedthe results inthe ChPTatO(p4).Explicitly, we find that F (p2 = 0) = 0.0945 (0.0425), 0.082 (0.034) and 0.106 (0.036) in V(A) the ChPT at O(p4), ChPT at O(p4) and LFQM, respectively. The differential decay rate as a function of x is given by dΓ m5 = K αG2 sin2θ A(x) (6) dx 64π2 F c 11 where the function of A(x) is given in Ref. . By integrating out the variable x in Eq. (6), in Table 1 we give the decay branching ratio of K+ e+ν γ. Here, as e → the IB term diverges at the limit of x 0 corresponding to p2 p2 = m2 , we → → max K have used the cuts of x = 0.01 and 0.1, respectively. With the cuts, from Table 1 we see that the IB contributions are much smaller than the SD± ones, which are insensitivetothecut.InFig.2,wealsodisplaythespectrumofthedifferentialdecay branching ratio in the ChPT at both O(p4) and O(p6) and the LFQM. From Fig. 2,we see that in the regionof x<0.7 or E <173 MeV, the decay branchingratio γ in the LFQMis much samllerthan that in the ChPT atO(p6). On the other hand, in the region of x > 0.7 the statement is reversed. However, if we only consider the contributions in the ChPT at O(p4), the conclusion is weaker. It is clear in 7 the future data analysis such as the one at the experiment BNL-E949 , one could concentrate on these two regions to find out which model is preferred. We have studied the axial-vector and vector form factors of the K+ γ tran- → sition in the LFQM and ChPT of O(p6). Based on these form factors, we have Fig.1. FV,A(p2)asfunctions ofthetransfermomentump2. February 2, 2008 16:46 WSPC/INSTRUCTION FILE Lih-ICFP07 4 Chen, Geng and Lih Table1. ThedecaybranchingratioofK+→e+νeγ (inunitsof10−5). Model xCut IB SD+ SD− Total ChPTatO(p4) 0.01 1.65×10−1 1.34 1.93×10−1 1.70 0.1 0.69×10−1 1.34 1.93×10−1 1.60 ChPTatO(p6) 0.01 1.65×10−1 1.15 2.58×10−1 1.57 0.1 0.69×10−1 1.15 2.58×10−1 1.47 LFQM 0.01 1.65×10−1 1.12 2.59×10−1 1.54 0.1 0.69×10−1 1.12 2.59×10−1 1.44 calculated the decay branching ratio of K+ e+ν γ. We have demonstrated that e → the SD parts give the dominant contributions to the decay in the whole allowed regionofthephotonenergyexceptthe lowendpoint.Future precisionexperimental 7 measurements on the decay spectrum should give us some useful information to determine the SD contributions as well as the vector and axial-vectorform factors. 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D56, 6856 (1997). 40 4 ChPT O(p) 35 ChPT O(p6) ) / dx 30 LFQM e 25 610 d Br ( K 112050 5 0 0.0 0.2 0.4 0.6 0.8 1.0 x (2E / mK) Fig.2. Thedifferentialdecay branchingratioasafunctionofx=2Eγ/mK.